Cantor Confusion
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From: MoeBlee on 13 Dec 2006 14:16 mueck...(a)rz.fh-augsburg.de wrote: > In *set theory* a set does not grow. In set theory a function is a set. > In set theory a function cannot grow. > In my view a function can grow. A function can be abbreviated as number > variable or as variable number. Notice the crucial difference: a theory vs. a view. MoeBlee
From: cbrown on 13 Dec 2006 14:25 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > > That one which goes left from the root is not engaged, because we need > > > > > only half of the set of edges. > > > > > > > > So your construction is not a surjection. > > > > > > Of course it is! It is a surjection from the set of edges onto the set > > > of paths. > > > > Than again. To what number maps the edge that goes left from the root? > > If it is a surjection, there should be one. > > > A surjection onto the paths covers all elements of the *range* = every > edge. What? The *range* of a surjection from edges onto paths is /not/ every edge; it is every path. The /domain/ of the surjection is every edge. (You actually teach Analysis, and you don't know this?!?!) > It is not necessary that every edge is mapped on a path, to show > that there are not less edges than paths. It is only necessary that > there is no path without edge mapped on it. > That is poorly stated. The mapping that maps the first edge to the left to every path provides us with a mapping such that "there is no path without edge mapped to it"; but this mapping does not prove that the cardinality of the set of paths <= the cardinality ({a single edge}) = 1. Cheers - Chas
From: Virgil on 13 Dec 2006 14:33 In article <1166007320.816114.309670(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > [...] > > >> 1. You do not present a convincing definition of "number". (Most > > >> likely you have none). > > > > > > Definitions are abbreviations like the following: > > > > [too long, too old, > > Impossible. Its from my new book to appear within few days. Can future > be too old? A new book with nothing in it but old ideas is not part of the future but merely an echo of the past. > > > too German; > > it is impossible to be too German. Only in the minds of Germans. > > > no definition at all] > > It is clear that you have not understood. On the contrary, it is likely that he understood too well.
From: Virgil on 13 Dec 2006 14:41 In article <1166007626.070058.215660(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > >> mueckenh(a)rz.fh-augsburg.de wrote: > > >> > Dik T. Winter schrieb: > > >> [...] > > >> >> Again you have provided neither a definition of "number", nor of > > >> >> "grow". > > >> >> Are you unable to do so? In common parlance, but that is not > > >> >> mathematics. In mathematics functions can grow in relation to > > >> >> their argument, but not the entities they denote. > > >> > > > >> > Functions cannot grow, according to modern mathematics. > > >> > > >> Wrong. I have provided a definition: > > >> > > >> ,----[ <45742128$0$97220$892e7fe2(a)authen.yellow.readfreenews.net> ] > > >> | Definition: A function f: A |-> B grows iff there exist a1 < a2 of > > >> | dom(f) and f(a1) < f(a2). We use the abbreviation "f grows" for of > > >> | "the function f grows". > > >> `---- > > > > > > The natural number n is a particular set of n elements. > > > If the number n can take a value n_1 and can take a value n_2 > > > with n_1 =/= n_2 then the number n can vary. > > > > How is that related to your sentence that "functions cannot grow"? > > It is the usual set theoretic view and as such in direct opposistion to > my view. It is the usual set theoretic view that numbers do not vary. If anything "varies" it is the value allowed for a variable, which can be any member of an unvarying set called its domain. > > I am glad to hear that many people read my texts. Repeated reading > supports understanding. But not acceptance. Repetition is a Baustein of learning. Repetition of falsehoods marks a Bausteinkopf.
From: Virgil on 13 Dec 2006 14:50
In article <1166008449.817555.138170(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > A_1 = {1} > > > > A_2 = {1,2} > > > > A_3 = {1,2,3} > > > > > > > > B = {1,2,3} > > > > > > > > then B is contained in the last A_i. If there is no last A_I, then > > > > there is > > > > no A_i that contains B > > > > > > That has nothing to do with "last". > > > > If A_i contains B, then A_i contains any A_j. > > Therefore A_i is "last". > > > > >It has all to do with "every A is finite". > > > > No. From the statement "every A is finite" we cannot conclude > > that there exists and A_i that contains B. > > That is correct. But this conclusion is derived from: > 1) N is a linear set (every finite initial segment (= line) includes > all preceding segments) > 2) There is no element of the complete segment (= diagonal) outside of > every finite segment (= line). But conclusion (2) requires the assumption that every set is finite, so the whole argument is circular and therefore invalid. > > Which of the following three do you claim > > (note that none of them require assuming that > > the set of natural numbers actually exists). > > > > - The potentially infinite sequence of > > natural numers is not an initial segment > > The potentially infinite sequence of natural numers is an initial > segment. (If we refrain from the physical constraints of numbers.) > > > > - there is a natural number which is > > not an element of the potentially infinite > > sequence of natural numbers > > No. Every natural number belongs to the sequence and to at least one > finite initial segment. > > > > - there is a line for which you cannot > > find a natural number which is not an > > element of the line > > Not so many nots please. > > For each line you can find a natural number which is not an element of > that line. And as soon as you have found (created) that number, you > have determined (created) the first line to which it belongs. > > There is no line with every natural number. > > Similarly you cannot find every elment of the diagonal. So that one can find any but not every? > If you name an > element, then the diagonal is constructed up to that element at least. And does it stop there? > > As there is no line with every number there can be no diagonal with > every number. That is an assumption which WM imposes, not a conclusion which can be drawn from anything incontrovertible. If WM will admit that he is taking it as an axiom, that is one thing, but he keeps insisting the he can deduce it from something which requires that he assume it. |